Abstracts of Talks

  • WHITNEY WORKSHOP
  • Published 2013

Abstract

S OF TALKS WHITNEY WORKSHOP, BANFF 2013, AS OF APRIL 24. Charles Fefferman. Finiteness Principle I, II, III. Abstract: These expository talks motivate the proof of the Brudnyi-Shvartsman finiteness principle for C(R) starting with the simplest nontrivial case, C(R), and then passing to the general case. Charles Fefferman. Whitney problems for Sobolev functions. Abstract: Extension of functions from finite subsets of R to functions in a Sobolev space. Theorems (proven and checked), algorithms (being written up, probably correct), open problems. Vladimir Goldshtein. Capacities in Sobolev spaces I, II. CANCELLED! Abstract: There are two main notions of capacities on Sobolev spaces: Sobolev capacity of compact sets and variational capacity of condensors. Sobolev capacity has found a number of applications in the theory of partial differential equations via the theory of Sobolev spaces. Variational capacity is a generalization of the conformal capacity and has found applications in the geometric function theory ( in the theory of quasiconformal mappings and its generalizations). We discuss the following subjects: Polar sets. A set is said to be polar if it has locally zero variational capacity. Polar sets are negligible sets for Sobolev spaces. Quasi-continuity (continuity outside of a set of arbitrary small capacity). A Lusin type theorem for the capacity says that every Sobolev function has a quasicontinuous representative. The property of quasicontinuity permits to redefine Sobolev spaces with the help of quasi-continuous representatives. Capacity as an outer measure. Choquet properties for the variational capacity. Existence of extremal functions and metric estimates for capacities. Capacitary properties of composition and extension operators. Vladimir Goldshtein. Quasiconformal Whitney Partition. CANCELLED! Abstract: A classical Whitney partition is a partition of a bounded Euclidean domain Ω into dyadic cubes with disjoint interiors, and with edges comparable to the distance to ∂Ω. Its modern generalization that is called a Whitney partition is a partition into convex polyhedra with uniformly bounded ratios of their exterior to interior radii, and with diameters comparable to the distance to ∂Ω. We propose a quasiconformal generalization of a Whitney partition where convex polyhedra are changed to quasiconformal images of balls under quasiconformal homeomorphisms of R onto itself, with uniformly bounded capacitory dilatations. We prove that the quasiconformal image of a quasiconformal Whitney family is a quasiconformal Whitney family. Capacitory estimates are the main tools in this study.

Cite this paper

@inproceedings{WORKSHOP2013AbstractsOT, title={Abstracts of Talks}, author={WHITNEY WORKSHOP}, year={2013} }